\begin{abstract}
We show tight necessary and sufficient conditions on the sizes of
small bipartite graphs whose union is a larger bipartite graph that
has no large bipartite independent set. Our main result is a common
generalization of two classical results in graph theory: the theorem
of K\H{o}v\'{a}ri, S\'{o}s and Tur\'{a}n on the minimum number of edges in
a bipartite graph that has no large independent set, and the theorem
of Hansel (also Katona and Szemer\'{e}di and Krichevskii) on the sum
of the sizes of bipartite graphs that can be used to construct a graph
(non-necessarily bipartite) that has no large independent set. Our
results unify the underlying combinatorial principles developed in the
proof of tight lower bounds for depth-two superconcentrators.
\end{abstract}
